Stability of Constrained Capillary Surfaces

Home , Capillary length, Capillary surface

FL47CH22-Steen ARI 1 December 2014 20:31

J.B. Bostwick1 and P.H. Steen2

1Department of Engineering Science and Applied Mathematics, Northwestern University, Evanston, Illinois 60208 2School of Chemical and Biomolecular Engineering and Center for Applied Mathematics, Cornell University, Ithaca, New York 14853; email: [email protected]

Annu. Rev. Fluid Mech. 2015. 47:539–68 Keywords First published online as a Review in Advance on surface tension, wetting, drops, bridges, rivulets, contact line, common line September 29, 2014

Annu. Rev. Fluid Mech. 2015.47:539-568. Downloaded from www.annualreviews.org The Annual Review of Fluid Mechanics is online at Abstract fluid.annualreviews.org A capillary surface is an interface between two fluids whose shape is deter- Access provided by Northwestern University - Evanston Campus on 01/06/15. For personal use only. This article’s doi: mined primarily by surface tension. Sessile drops, liquid bridges, rivulets, and 10.1146/annurev-fluid-010814-013626 liquid drops on fibers are all examples of capillary shapes influenced by con- Copyright c 2015 by Annual Reviews. tact with a solid. Capillary shapes can reconfigure spontaneously or exhibit All rights reserved natural oscillations, reflecting static or dynamic instabilities, respectively. Both instabilities are related, and a review of static stability precedes the dynamic case. The focus of the dynamic case here is the hydrodynamic stability of capillary surfaces subject to constraints of (a) volume conservation, (b) contact-line boundary conditions, and (c) the geometry of the supporting surface.

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1. OVERVIEW Capillary surfaces are at once a modern and classical topic. Shape reconfiguration is central to Capillary surface: a emerging applications that exploit the physics of liquids confined by surface tension on the micro- mathematical surface and nanoscale, whereas the study of shape stability reaches back to the 1800s (see the sidebar between two fluids endowed with surface Historical Perspective). We review recent progress in the dynamic stability analysis of surfaces, tension focusing on the influence of constraint by solid supports. Volume (pressure) An early stability prediction is due to Plateau (1863), who reported the instability to volume dis- disturbance: turbances of a capillary cylinder for lengths longer than its circumference, the well-known Plateau disturbance that limit. Some years later, Rayleigh (1879) calculated the growth rate of instability as it depends on preserves the volume the disturbance wave number to estimate the final droplet size for a liquid jet disintegrating into (pressure) of the base droplets. The Plateau limit is recovered from Rayleigh’s solution of the hydrodynamic problem by state putting the growth rate to zero [the Plateau limit is sometimes referred to as the Plateau-Rayleigh Plateau-Rayleigh (PR) limit]. This illustrates how static stability can be recovered from a dynamic analysis and (PR) limit: the length, equal to its provides the prototypical relationship between the stabilities, a theme of this review. base-state Surface tension largely determines the surface shape on scales smaller than the capillary length 1/2 circumference, beyond lc ≡ (σ/ρg) ,whereρg is the buoyant force per volume according to the gravity level g and the which a cylindrical density difference between the fluids ρ. For fluids separated by a thin film (e.g., soap films) or a capillary surface is fluid against an immiscible fluid (e.g., Plateau tanks), the density difference ρ can be small, yielding unstable to volume disturbances tens-of-centimeter capillary lengths. In space applications, the capillary length for a liquid against gas can be 100 cm, whereas it is usually a few millimeters on Earth. Plateau tank: a chamber that enables Michael (1981) gave a then-timely snapshot of the field in this journal, with an emphasis large lc on Earth by on static stability and methods. Interest in shape stability, with its enhanced role in low-gravity immersing a capillary fluid mechanics, grew as space programs grew. Myshkis et al. (1987) provided a comprehensive surface in an monograph with this motivation. Also informed by capillarity in weightlessness is Langbein (2002). immiscible liquid of De Gennes (1985) paid particular attention to wetting and spreading, which culminated in a nearly the same density comprehensive account of capillary phenomena (de Gennes et al. 2010). Rowlinson & Widom Sessile drop: a liquid (2002) and others in the chemistry community provided a molecular perspective, and yet another drop on a solid substrate thread, with mathematical roots in the study of isoperimetric inequalities, has been provided by Finn (1999). Johns & Narayanan (2002) presented case studies in capillary stability, whereas Liquid bridge: a liquid enclosed by a Shikhmurzaev (2007) focused more on the flows that underlie capillary surfaces. In this article, we capillary surface focus on the mechanics of fluids subject to capillary instability. having two closed- Between 1994 and 2013, items published annually that list the terms sessile drop, liquid curve contact lines bridge, or liquid rivulet as a topic have surged nearly sixfold in number (Figure 1a). Nearly 3,000 located on separate publications have appeared in total. Using the terms capillarity, wetting, or spreading as alternative solid surfaces Annu. Rev. Fluid Mech. 2015.47:539-568. Downloaded from www.annualreviews.org

HISTORICAL PERSPECTIVE Access provided by Northwestern University - Evanston Campus on 01/06/15. For personal use only.

Popular public lectures by Boys in the 1890s featured soap-film experiments. Boys (1959 [1890]) attributed these experiments in capillarity to the published work of Savart, Plateau, Maxwell, Thomson, Rayleigh, Bell, and Rucker.¨ An entry on capillary action first appeared in the Encyclopedia Britannica in 1898 (Maxwell 1898), reflecting the mathematical physicists’ wide interest in the subject at that time (e.g., Schrodinger¨ 1915). About one-third of the 14-page entry speaks to the stability of capillary surfaces. The entry remains in the encyclopedia until 1926, at which time it redirects readers to an entry on surface tension, with an indication that Lord Rayleigh has taken over authorship, at least for the historical review part. Regarding the 1898 entry, Erle et al. (1970) pointed out that Maxwell confused volume and pressure disturbances in his discussion of the stability of the catenoid.

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240 220 abc 200 180 Search topic V Liquid/fluid bridge 160 Sessile drop/droplet 140 Df Γ 120 ( ) 100 D(V) 80 Liquid σ 60 α

Number of published items σ σ 40 ls sg 20 Ds Solid 0 p 1994 1998 2002 2006 2010 2013 Calendar year

Figure 1 (a) Number of publications by year since 1994 found by searching on listed topics. (b) Deﬁnition sketch for a sessile drop of volume V α σ σ σ and equilibrium contact-angle , with liquid/gas ( ), liquid/solid ( ls), and solid/gas ( sg) interfacial energies per area. (c) Sessile drop response diagram for a pinned contact line: volume V against Laplace pressure p.

topics shows an even stronger swell. Feeding this are streams of interest from new applications, mainly on the micro- and nanoscales. For droplets, these applications include nanoparticle assembly by droplet evaporation (Bigioni et al. 2006), the assembly of microparts via capillarity (Sariola et al. 2010), microassays for screening (Berthier et al. 2008), microencapsulation (Almeida et al. 2008), forensic bloodstains (Attinger et al. 2013), immersion lithography at high speeds (Harder et al. 2008), the cleaning of nanopatterned wafers (Xu et al. 2013, Shahraz et al. 2012), biomimetic surface design (Lafuma & Quer´ e´ 2003, Tuteja et al. 2007), and surface wettability engineering generally. Contact-angle measurement continues to be a principal means for the assessment of material wettability in many industries (Matsumoto & Nogi 2008, Seetharaman et al. 2013). For liquid bridges, applications include gravure printing (Dodds et al. 2012, Kumar 2015), colloidal aggregation (Kralchevsky & Denkov 2001), solid adhesion (Maugis 2000), bioinspired adhesion (De Souza et al. 2008, Slater et al. 2012, van Lengerich & Steen 2012), crystal growth in ﬂoat zones (Lappa 2005), and extensional rheology (McKinley & Sridhar 2002), whereas, for rivulets, they include heat and mass transfer (El-Genk & Saber 2001). Inexpensive Contact angle: angle

Annu. Rev. Fluid Mech. 2015.47:539-568. Downloaded from www.annualreviews.org access to high-speed imaging cameras may also be playing a role in the swell of interest. measured through a Simulations (Sui et al. 2014) and experiments (Milne et al. 2014) are crucial to the subject’s liquid, from the solid to the capillary surface; Access provided by Northwestern University - Evanston Campus on 01/06/15. For personal use only. development. Related, but also outside the scope of the present review, are thin films and coating a static or equilibrium flows (Oron et al. 1997); Marangoni flows (Matar & Craster 2009); the nonlinear dynamics of contact angle relates topological change that sometimes ensues from capillary instability (Eggers 1997, Paulsen et al. the liquid/gas, 2012); and software such as Surface Evolver (Brakke 1992), a versatile tool for predicting the liquid/solid, and reconfiguration of static surfaces (Collicott & Weislogel 2004). solid/gas surface Liquids that partially wet a solid support are of interest (Figure 1b). For these, liquid-gas (A ), energies through the lg Young-Dupre´ liquid-solid (Als), and solid-gas (Asg) surface areas can all change upon shape reconfiguration. equation ThefreeenergyU of a configuration is given by the sum of its surface areas A weighted by the σ Rivulet: a liquid corresponding interfacial energies per area (equivalently, surface tensions) , stream on a solid substrate U = σ Alg + σls Als + σsg Asg. (1)

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Dry solid appears at the same rate that liquid recedes, δ Asg =−δ Als, enabling a reduced energy functional to be defined, A ≡ − M , Contact line (CL): Alg Als (2) the line intersection of where M ≡ (σsg − σls)/σ is the relative affinity of the partially wetting liquid for the solid, with a capillary surface with −1 ≤ M ≤ 1. If material properties are such that M falls outside (−1, 1), the liquid completely a solid support surface; wets (M > 1) or dewets (M < −1) the solid. common to liquid, gas, and solid phases Stability determinations are sensitive to the class of allowable disturbances. Perhaps the greatest sensitivity is to fixed (pinned) or moving (free) contact lines (CLs), as these control whether one or Fold: a turning point two competitors are active in the competition to decrease A. For pinned CLs (δ Als = 0), only the Response diagram: liquid-gas area can change, δA = δ Alg, whereas for free CLs, both δ Alg and δ Als can be of either a plot of force-like δA = response against sign. The overall competition, determined for equilibrium by 0 and for stability (sufficient) 2 deflection-like by δ A > 0, depends on the weight M, which can have either sign, depending on whether the displacement, as in a liquid is wetting, cos α>0, or nonwetting, cos α<0, as the equilibrium contact angle α is related, pressure-volume cos α = M (Section 2.2). diagram Stability is also sensitive to constraints on the bulk—whether pressure or volume disturbances

are allowed. Let us consider a surface area Alg enclosing a liquid volume V so that Alg depends on V alone, as for the sessile drop in Figure 1c with V controlled by injection through the base, for example. For a large class of shapes, including those of constant mean curvature with pinned CL, it has been shown that dAlg p = , (3) dV σ where p(V) is the capillary pressure difference across the surface (Gauss 1830, Gillette & Dyson 1974). According to Equation 3, equilibrium (dAlg/dV = 0) requires p = 0, and for stability 2 2 (d Alg/dV > 0), dp/dV > 0. We note that this calculation does not account for any constraint on the disturbance volume. The criterion for stability, dp/dV > 0, turns out to be sufficient but not necessary. A thought experiment illustrates (see Figure 1c). For a disturbance at constant pressure (referred to as a pressure disturbance) that adds volume, when dp/dV > 0, the drop in equilibrium responds with greater pressure, expelling the newly added volume and falling back to equilibrium. This is stability. When dp/dV < 0, the opposite occurs, giving instability. That is, below (above) the fold in the pV curve, the sessile drop is stable (unstable). Alternatively, for a disturbance that preserves the volume, a volume disturbance, the entire family of equilibria is stable. In summary, below the fold, the equilibria are stable to both disturbances, whereas, above the fold, they are stable to volume but not to pressure disturbances. To obtain this stability result by calculus of the second variation from Equations 2 and 3, one must account for the volume constraint. Poincare´ (1885) provided an approach that circumvents calculation of the second variation, δ2A, giving both pressure and volume Annu. Rev. Fluid Mech. 2015.47:539-568. Downloaded from www.annualreviews.org stability changes from folds in the response diagram, which consists of a family of equilibria (see Section 2.4). Access provided by Northwestern University - Evanston Campus on 01/06/15. For personal use only. Reconfiguration by surface tension occurs dynamically at speeds u ≡ (σ/ρ)1/2, which can be used to define an inverse Reynolds number, the Ohnesorge number, Oh ≡ μ/(ρσ)1/2,whereμ is the dynamic viscosity. We note that the reconfiguration speed u increases with decreasing scale .For ∼ 0.1 mm and water in air, one obtains Oh ∼ 10−2. Basaran (2002) gave examples of inviscid behavior at scales well below 1 mm. Low–Ohnesorge number flows are our interest here. In summary, our goal is to illustrate the mathematical theory of capillary instability using relevant examples to guide the reader to the forefront of this exciting field. The challenge is to develop a perspective that yields rules of thumb for the practitioner. Common configurations are given in Figure 2d–f. The far-left column of Figure 2 highlights sessile supports of varying primary curvatures, whereas the far-right column highlights varying

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SESSILE DROPS LIQUID BRIDGES IMPALED DROPS

acb

Liquid

Solid

Concave support Concave support Concave support

dfe

Planar support Planar support

Cylindrical support

gih

Convex support Convex support Convex support Pendular ring

Figure 2 Spherical surface configurations with axisymmetric support geometry. Annu. Rev. Fluid Mech. 2015.47:539-568. Downloaded from www.annualreviews.org secondary curvatures. The middle column of the figure shows liquid bridge configurations. When a dome support is part of a bridge, the configuration is a pendular ring (Figure 2h), and a latitudinal Access provided by Northwestern University - Evanston Campus on 01/06/15. For personal use only. belt is referred to as a spherical belt (Figure 2b). Homotopy: The columns in Figure 2 show families that can be related by the support variation. As such, a continuous map the stability results for one family member can often be inferred from those of another. Canonical relating one support support geometries represent each column: the sessile drop, liquid bridge, and impaled drop geometry to another (Figure 2d–f, respectively). A sessile drop is related to a free drop and a liquid bridge to a cylindrical and, thereby, one bridge, and both are related through a disintegrating jet, as discussed above. In addition to their system of equations and boundary organizational utility, such familial relationships, mathematically formalized through problem conditions (or homotopy, can be useful for quantitative predictions (see Section 3.2.2). problem) to another Figure 3 illustrates influences of solid support on static stability. Helical supports enable long liquid columns. Figure 3a shows a liquid filling a stretched spring and the column near

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a bc

iii

Figure 3 Photographs of static shapes in Plateau tanks. (a) Liquid ﬁlling a stretched spring (i ) and the liquid column near its (ii ) minimum-volume and (iii ) maximum-volume stability limits. Panel a reproduced with permission from Lowry & Thiessen (2007). (b) Flow control revealing the pearl-shaped mode, the next static instability beyond the Plateau-Rayleigh limit (Lowry & Steen 1997). Photograph courtesy of Brian Lowry. (c) Beer-belly shape observed when an impaled drop (liquid bridge) is squeezed by end supports at a ﬁxed volume, typical of the azimuthal instability seen when the Steiner limit is exceeded. Panel c reproduced with permission from Russo & Steen (1986).

its minimum-volume and maximum-volume stability limits. Figure 3b shows a flow-influenced PR instability. After an unconstrained PR instability, an amphora shape appears while here an annular external flow stabilizes the first instability mode so that the surface collapses at the second instability mode, exhibiting a pearl-shaped configuration. In Figure 3c, on decreasing length at a fixed volume, the impaled drop (or liquid bridge) suffers an azimuthal instability at the Steiner limit. Configurational equilibrium requires bulk, surface, and CL equilibrium. Bulk equilibrium requires an equilibrated hydrostatic pressure field throughout the liquid. Surface equilibrium requires the pressure at the liquid surface to match the Young-Laplace pressure. Finally, CL equilibrium requires either a fixed CL (the pinned condition) or a mobile CL with a fixed equilibrium contact angle (the free condition), as required by the Young-Dupre´ equation. Provided gravity is neglected, all configurations in Figure 2 are in bulk and surface equilibrium Annu. Rev. Fluid Mech. 2015.47:539-568. Downloaded from www.annualreviews.org as each shape is a spherical section. Solid properties must be accommodating for CL equilibrium to obtain. That is, the solid supports must be arranged, via local geometry (sharp corners) or chemistry Access provided by Northwestern University - Evanston Campus on 01/06/15. For personal use only. (material properties), to yield a pinned or free CL. For the pinned CL, all the capillary surfaces of Figure 2 are statically stable. Stability is an immediate consequence of (a) the local minimum of the surface energy of the spherical shape and (b) the restriction of the class of disturbances caused by pinning the CL(s). That is, if the free sphere is stable locally (isoperimetric inequality), a section of a Azimuthal mode or sphere with a pinned boundary must also be stable, as it is subject to fewer disturbances. In contrast, shape: mode or if the CL is free, the capillary figures may be unstable, even if they are in CL equilibrium. A free disturbance shape that CL brings the surface energy of the solid into play, adding a degree of freedom to the competition breaks rotational symmetry that can be destabilizing. For example, for a thick wire, the impaled drop in the far-right column of Figure 2 can lower the liquid-solid-gas system energy by shifting to the perimeter of the wire, breaking axisymmetry, and becoming more like an isolated drop (Quer´ e´ 1999).

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In summary, when a solid is introduced to support a drop, the free surface becomes constrained. This is stabilizing because it narrows the class of disturbances. Conversely, if the CL is free to wet or dewet the solid, a new freedom arises, which is destabilizing. The competition between Turning point (TP): the stabilizing and destabilizing effect is our subject. For dynamic stability, in addition to these a point in the preferred complications, the solid support introduces the laboratory frame, which can introduce a low- diagram at which the frequency oscillation mode. We now discuss progress in resolving this competition, first for the response curve has a static and then for the dynamic disturbances. horizontal or vertical tangent, turning back without crossing 2. HISTORICAL FOUNDATIONS AND STATIC STABILITY Interest in capillary phenomena began in the 1700s when Segner (1751) first postulated the concept of surface tension. Young (1805) and Laplace (1806) made the subject quantitative. Whereas Young preferred to use forces, Gauss (1830) promoted the energy-minimization approach. This required the calculus of variations. The equations governing the extremals and minimizers for supported capillary shapes have been known for more than 100 years (Bolza 1904). We record them here to establish the parallel formalism with the dynamic problem, using the notation of Myshkis et al. (1987). We then discuss alternate solution methods to the direct calculation of the second variation, such as the Poincare´ turning-point (TP) method, and place them in a historical context (see the sidebar Rotating, Self-Gravitating Figures of Equilibrium). Finally, we briefly review the stability of systems of capillary elements, including capillary switches and compound drop and bridge systems.

2.1. Energy Functional A liquid that partially wets a solid substrate has potential energy U given by Equation 1. In many situations, the volume V enclosed by the capillary surface is held constant: V [x] ≡ dV = C [D]. (4) V

ROTATING, SELF-GRAVITATING FIGURES OF EQUILIBRIUM

The shape of a rotating self-gravitating liquid body was of sustained mathematical interest for more than a century, beginning with Newton’s prediction of Earth being a sphere ﬂattened at the poles. This model of planetary evolution

Annu. Rev. Fluid Mech. 2015.47:539-568. Downloaded from www.annualreviews.org depends on the gravitational strength and rotational rate. Two-parameter families of equilibrium shapes and their stability were studied by Maclaurin, Jacobi, Meyer, Liouville, Dirichlet, Dedekind, and Riemann, among others.

Access provided by Northwestern University - Evanston Campus on 01/06/15. For personal use only. Chandrasekhar (1987) provided a historical account of their efforts, along with the many missteps that were made. As part of the discussion at the time, Poincare´ (1885) showed that TPs along solution branches signal stability changes and thereby provided an economical way to track stability. For these historical reasons, the astrophysics community has probably had a greater awareness of Poincare’s´ approach (Lynden-Bell & Wood 1968; Katz 1978, 1979) than has the mechanics community, whose awareness is growing (Thompson 1979, Maddocks 1987, Lowry & Steen 1995, Luzzatto-Fegiz & Williamson 2012). Brown & Scriven (1980b) considered the shape and stability of rotating drops held by surface tension using the ﬁnite-element computational approach and, perhaps unsurprisingly, found numerous branching families of equilibria reminiscent of those summarized by Chandrasekhar (1987). Brown & Scriven (1980b) computed stability by direct evaluation of the second variation.

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x x x ab Γ Df(k)

s n Liquid Liquid

D D δx = yn γ n1 α

Ds(k)

Equilibrium Disturbed

Figure 4 Region near contact line γ for a partially wetting liquid showing the (a) equilibrium (blue solid line) x¯ and (b) disturbed (blue solid line) x = x¯ + δx conﬁgurations with δx = yn. The capillary surface , with normal curvature k and surface normal n, intersects the solid s support ∂ D ,withnormalcurvaturek¯ and surface normal n1, at the contact line γ to make the contact angle α.

One can treat the volume-conservation constraint as an auxiliary condition by introducing the Lagrange multiplier μ into an augmented functional, F[x] = U[x] − μV [x]. (5)

2.2. Equilibrium (First-Order Conditions): Bulk, Surface, and Contact Line For motionless base states, the pressure must be hydrostatic in the bulk. For negligible gravity, this implies a uniform pressure throughout. The surface equilibrium conditions arise from the first variation of the energy functional. The capillary surface is perturbed, x = x¯ + yn (see Figure 4), and applied to the energy functional (Equation 5). The first variation requires that equilibria x¯ satisfy the first-order conditions 1 y δF[x] = (κ1 + κ2 − μ) yd + (n · n1 − cos α) dγ (6) σ γ sin α for all allowable disturbances y. Here we note that the first-order conditions for the augmented functional δF = 0 are the same as those for the energy functional δU = 0 because of the volume- Annu. Rev. Fluid Mech. 2015.47:539-568. Downloaded from www.annualreviews.org conservation constraint δV = 0. A vanishing first variation, δF = 0, yields the equilibrium condition for the liquid/gas interface Access provided by Northwestern University - Evanston Campus on 01/06/15. For personal use only. , p = ≡ κ + κ ∂ f , σ 2H 1 2 [ D ] (7)

which relates the pressure p to the principal curvatures κ1 and κ2 there (Young 1805, Laplace 1806). For negligible gravity, Equation 7 implies that equilibrium surfaces have constant mean curvature H and belong to one-parameter families (see the sidebar Surfaces of Constant Mean Curvature). Similarly, requiring δF = 0 delivers two alternative equilibrium conditions for the CL. The ﬁrst is simply the pinned CL equation, y = 0[γ ], (8)

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SURFACES OF CONSTANT MEAN CURVATURE

Surfaces of constant mean curvature can be bounded or unbounded, singly or multiply periodic, self-intersecting or not. Delaunay (1841) gave a geometrical construction for all surfaces of revolution of constant mean curvature, classified by their curve of generation. These shapes are nodoids and unduloids, along with the special cases of the sphere, cylinder, and catenoid (Gillette & Dyson 1971). Along with the plane, these shapes exhaust the axisymmetric solutions to Equation 7 with constant H. Pieces of these shapes also constitute solutions. In addition, there are many unbounded constant H shapes without rotational symmetry. Block copolymers have been observed to undergo morphological changes between constant H configurations (Thomas et al. 1988, Jain et al. 2005). DNA structure motivates the study of the helicoid to catenoid (H = 0) transition, tested using soap films on a wire frame (Boudaoud et al. 1999). May & Lowry (2008) obtained the stability limits of volumes supported by dual helical boundaries, extending the work of Lowry & Thiessen (2007).

and the second is a geometric condition,

n · n1 − cos α = 0[γ ], (9)

relating the surface normals n and n1. Equation 9 is referred to as the natural boundary condition in the calculus of variations. Substituting cos α = M into Equation 9 yields the Young-Dupre´ equation in the surface-normal form (Young 1805, Dupre´ 1869).

2.3. Stability (Second-Order Conditions): Disturbance Classes The stability of the equilibrium surface x¯ is determined by solving the eigenvalue problem associated with the second variation δ2 ≡− − κ2 + κ2 − μ = λ ∂ f . F[y] [y] ( 1 2 ) y y y [ D ] (10)

Here y is the interface disturbance, and is the Laplace-Beltrami operator or surface Laplacian, which is deﬁned on the equilibrium surface, ∂ √ ∂ 1 ij y f [y] ≡ √ gg , g ≡ x¯, ·x¯, , i, j = 1, 2[∂ D ], (11) g ∂ui ∂u j ij i j

where gij is the surface metric tensor, using notation standard to differential geometry. Eigenvalues Annu. Rev. Fluid Mech. 2015.47:539-568. Downloaded from www.annualreviews.org can be shown to be real (self-adjoint operator). The sign of the eigenvalue λ gives the stability of a particular mode; λ>0 implies that the equilibrium is stable to that disturbance mode, whereas Access provided by Northwestern University - Evanston Campus on 01/06/15. For personal use only. λ < 0 corresponds to instability. For the equilibrium to be stable, there can be no unstable modes. We note that, in the absence of symmetry, Equation 10 is a partial differential equation because disturbances take the form y(u1, u2), where u1 and u2 are the surface coordinates. Constraints on the bulk and at the CL inﬂuence stability, as mentioned in Section 1. For the bulk, one ﬁnds that either μ = 0, corresponding to disturbances that preserve the volume, called volume disturbances, or μ = 0, corresponding to disturbances of constant pressure, called pressure disturbances. For volume disturbances, one augments the second variation (Equation 10) with the following auxiliary condition on the function y to determine μ: yd = 0[∂ D f ]. (12)

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PINNING CONTACT LINES IN PRACTICE

Pinning CLs in practice requires some combination of chemical and geometrical strategies. Several methods, some well based in prediction, are now part of the spacecraft engineer’s toolbox. The most commonly used way to locate CLs under conditions of weightlessness is to introduce sharp corners (Langbein 2002, chapter 7), often by joining wedges of different materials to form the corner. The sharpness of the edges (the wedge angle) must satisfy speciﬁc conditions relative to the respective contact angles to be effective. Gibbs (1906) considered this problem and derived inequalities that hold when the CL coincides with a sharp edge of a solid support. Dyson (1988) raised objections to Gibbs’s proof and provided a counterexample.

At the CL, for both volume and pressure disturbances, consistency with ﬁrst-order conditions (Equation 6) requires either (a) a ﬁxed CL (pinned CL) (Myshkis et al. 1987),

y = 0[γ ], (13)

or (b) a geometric condition (free CL),

y + χy = 0,χ≡ (k cot α − k¯/ sin α)[γ ], (14)

where χ is a parameter related to the static contact angle α, the normal curvature k of the free surface, and the normal curvature k¯ of the solid support (Figure 4a). Disturbances to the free CL can behave as though pinned, χ →±∞, or as preserving α, χ = 0. In this way, one may think of χ as characterizing how easily a disturbance can be anchored, and Equation 14 suggests how chemical α and geometrical k¯ modiﬁcations of a support surface compete or cooperate to pin a CL. In many applications, CL pinning is crucial, but doing so in practice remains largely an art (see the sidebar Pinning Contact Lines in Practice). We discuss how the geometry of the solid support affects stability in Section 3.2.2. Direct computation of the spectrum λ of Equation 10 becomes challenging when it depends on multiple parameters. For volume disturbances μ = 0, the computation is also complicated because one must enforce the auxiliary condition given in Equation 12. In this case, the Lagrange multiplier μ is treated as an additional parameter that must be determined as part of the solution, effectively increasing the system size. For example, Slobozhanin et al. (1997) and Myshkis et al. (1987) employed direct calculation to identify the critical disturbance for the liquid bridge with pinned CLs. Annu. Rev. Fluid Mech. 2015.47:539-568. Downloaded from www.annualreviews.org The conjugate-point method also directly computes stability. The focus is on the least-stable direction and its possible destabilization. The approach involves computing conjugate points of Access provided by Northwestern University - Evanston Campus on 01/06/15. For personal use only. the Jacobi equation [the second variation (Equation 10) with λ = 0]. To prove stability, one must (a) satisfy Legendre’s condition and (b) prove the absence of a conjugate point (Bolza 1904).

Poincare-Maddocks´ (PM) method: 2.4. Poincare´ Turning-Point Method stability changes are inferred from turning An alternative to direct computation is a TP method, which goes back to Poincare´ (1885), with a points and bifurcations modern treatment by Maddocks (1987), sometimes referred to as the Poincare-Maddocks´ (PM) in the preferred method. This bifurcation-theoretic approach extracts information from families of equilibria and diagram thereby avoids solving Equation 10 directly. The advantage is considerable if one is primarily interested in changes in stability. The method is well suited to the isoperimetric nature of capillary

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INDEX THEORY FOR ISOPERIMETRIC PROBLEMS

Underlying the PM theory is an index theory, related to the number of negative eigenvalues of the second variation on a constrained function space S, that codiﬁes the relationship between pressure and volume disturbances. The central idea is that the unconstrained (pressure) index [U] is simply related to the constrained (volume) index [C] through the following relationship, [U] = [C] + [C⊥], where [C⊥] is the index on the orthogonal complement to the constrained function space S⊥, which is related to the curvature of the augmented functional ∂2 F/∂λ2 (Maddocks & Sachs 1995). The main PM theorem then follows immediately because the curvature, by deﬁnition, changes at a fold in the preferred diagram. Volume disturbances are relatively stable to pressure disturbances, as exhibited by the relationship between the constrained and unconstrained index.

surface problems (Lowry & Steen 1995) and to computational continuation (Doedel & Oldeman 2009). Preferred diagram: We provide an example using the preferred diagram for a liquid bridge with equal pinned a response diagram, CLs (Figure 5). This family of equilibria, all pieces of Delaunay shapes, has been obtained by according to the continuation in arc length along the branch, starting from the cylindrical bridge, just below B, Poincare-Maddocks´ and following in one direction and then reversing along the other direction. Three TPs appear: prescription, that has two extremals in pressure, between shapes A and B and between shapes B and C, and one in theformatofthe λ isoperimetric volume, between shapes C and D. PM theory states that the number of unstable eigenvalues (constrained) variable (solutions to Equation 10) changes at TPs. Speciﬁcally, the stability index changes at the pressure against the Lagrange TPs for pressure disturbances and at volume TPs for volume disturbances (see the sidebar Index multiplier (constraint) Theory for Isoperimetric Problems). Because a cylinder of length L = 2.72 is stable to both and consists of a family pressure and volume disturbances (Plateau 1863), the starting point has indices ([p], [V]) = (0, of equilibria 0). On decreasing volume, a pressure TP is ﬁrst traversed to make the bridge pressure unstable Stability index: the ([p], [V]) = (1, 0) until beyond shape C, where the volume TP is then traversed and equilibria number of negative = eigenvalues of the become volume unstable. Shape D is both pressure and volume unstable ([p], [V]) (1, 1). In the second variation when other direction, index changes are similarly read off the response diagram. Using PM theory to infer restricted to a stability changes is only useful when rotationally symmetric disturbances are most dangerous. This particular disturbance can sometimes be proved by Steiner symmetrization. That is, when symmetrization is possible, class; an index 0 symmetric disturbances are most dangerous. The limiting shape at which the procedure ceases to represents a stable equilibrium work is the Steiner limit (see the sidebar Steiner Symmetrization).

STEINER SYMMETRIZATION Annu. Rev. Fluid Mech. 2015.47:539-568. Downloaded from www.annualreviews.org

Steiner (1882) introduced a geometric procedure to map a three-dimensional shape onto a rotationally symmetric Access provided by Northwestern University - Evanston Campus on 01/06/15. For personal use only. shape, preserving the volume while not increasing the surface area, which Gillette & Dyson (1972) proved for bridges with pinned CLs. This mapping procedure works, as long as the rotationally symmetric shape (and its three- dimensional disturbance) is single-valued in the radial coordinate of the cylindrical system. The limiting symmetric shape for a liquid bridge is therefore one that comes in tangent to the CL, which we call the Steiner limit. When constructible, the mapping proves that symmetric disturbances are most dangerous. When not constructible, there is no information. However, at least in the case of a liquid bridge, the Steiner limit also corresponds to the instability to azimuthal disturbances (Figure 3c). Slobozhanin et al. (1997) proved this by solving Equation 10 at the Steiner limit. When symmetric disturbances are most dangerous, the second variation reduces to an ordinary differential equation for the function y(u), a great simpliﬁcation.

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100 ([p], [V]) (1, 0) 80 (0, 0) (1, 0) (1, 1)

40 A (scaled) V

Turning 20 point Volume, Turning B point C D 0 Turning point

–20

0.8 1.0 1.2 1.4 1.6 1.8 Pressure, p (scaled)

Figure 5 Pressure-volume response of a liquid bridge with a pinned contact line of scaled length L = 2.72. Rendered equilibrium shapes are labeled A–D. Stability changes, indicated by stability indices [p]and[V], occur at turning points ( yellow circles) in pressure (between A and B and between B and C) and in volume (between C and D).

2.5. Systems: Compound Drops and Droplet Switches Nowadays, double bridges, double drops, and other capillary systems are of interest for device design. Classically, they were of interest to test theory (Rucker¨ 1886). In this section, using droplet systems, we illustrate the relevance of stability to design and, in the next, using compound bridges, we show how the PM method can be adapted to predict system stability. Figure 6a illustrates a system of water drops undergoing a slow coarsening process. In the system is an array of 24 holes, drilled in a plate, with a reservoir attached beneath. Through a Annu. Rev. Fluid Mech. 2015.47:539-568. Downloaded from www.annualreviews.org valve, a syringe overﬁlls the reservoir until the 24 drops protrude as superhemispheres (t = 0s in Figure 6a). The valve is then closed so that the system, the drops and reservoir, has a ﬁxed

Access provided by Northwestern University - Evanston Campus on 01/06/15. For personal use only. total volume. The initial state is unstable. As time progresses, owing to slightly varying initial curvatures (hence pressures), ﬂuid is pumped to bigger drops, and the volume collects increasingly into fewer drops. After long-enough time, all excess volume ends up in a single drop (t = 40 s in Figure 6a). The system has fully coarsened to an equilibrium state (neglecting evaporation). According to the theory of single-drop response (Figure 1c), were they unconstrained, all n drops initially would be unstable as they fall on the upper branch. However, they must obey the overall volume constraint. Thus, the system may be thought of as having n − 1 rather than n degrees of freedom. The ﬁnal rest state after 40 s in Figure 6a shows that 23 drops have behaved unstably, as if the stability were to pressure disturbances (unconstrained), whereas the big drop has behaved stably, as if responding to volume disturbances.

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ab0 s d0 ms

2 s

iivii iii

21 s 76 ms

40 s

Figure 6 Dynamic instability. (a) Video images showing coupled drops coarsening in time, starting as 24 big drops of nearly equipartitioned volume (unstable equilibrium), t = 0 s, relaxing to six big drops at 2 s, to two big drops at 21 s, and to one big drop at 40 s, the final (stable) equilibrium. Panel a reproduced with permission from van Lengerich et al. (2010). (b) Time-sequence photos showing capillary switch (side view) toggling between droplet-droplet (i, iv) and droplet-bridge (ii, iii ) configurations. Photographs courtesy of A. Hirsa. (c) Sessile drop (top view), mechanically excited (megahertz; Oh ≤ 0.003), exhibiting (left) circular standing waves that, upon being shaken harder, lose symmetry to (right) asymmetric surface patterns. Panel c reproduced with permission from Vukasinovic et al. (2007). (d ) Sessile drop (top view through a fixed mesh), mechanically excited (kilohertz; Oh = 0.0024), exhibiting a resonant mode [5, 5] at two instants one half period apart. Panel d reproduced with permission from Chang et al. (2013).

Regarding the coarsening of coupled water drops, a theory that predicts the instability of the initial state and the stability of the final state is available (Gillette & Dyson 1974, Slobozhanin & Alexander 2003). More generally, the theory predicts how the static stability of a system depends on the element stability. For n elements coupled in parallel with a system volume constraint = n / > = ,... V i V i , having all elements stable, (dp dV )i 0, for i 1 n, is sufficient for system stability but not necessary. Necessary conditions allow at most one element to take a configuration that would otherwise be unstable (i.e., as if uncoupled). Only under a certain additional condition do the necessary conditions become sufficient for system stability. This additional condition has been proved using a direct calculation of the second variation for coarsening of n drops coupled Annu. Rev. Fluid Mech. 2015.47:539-568. Downloaded from www.annualreviews.org by various network topologies (van Lengerich et al. 2010), with relevance to the experiment just recounted. The related three-coupled drop instability has been treated by direct calculation from Access provided by Northwestern University - Evanston Campus on 01/06/15. For personal use only. a catastrophe-theory perspective (Wente 1999). Another example of the system behavior is provided by compounding two drops to form a double-welled energy landscape (Boys 1959 [1890]). When a means to toggle between the bistable states is introduced, a capillary switch is realized. Various toggling activations have been introduced, including acoustic (Hirsa et al. 2005), electro-osmotic (Vogel et al. 2005, Barz & Steen 2013), magnetic (Malouin et al. 2010), and electric fields (Sambath & Basaran 2014). Figure 6b shows a 1-mm-thick plate in which a single hole has been drilled and then overfilled with fluorescein-laden water, illuminated by a laser sheet. In the static configuration (part i), pinned drops protrude above and below the plate, and the bistable state big-down is shown. Parts ii–iv in Figure 6b show snapshots of the switch during toggling against a second plate,

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3.0 Az Rotund D 2.5 Rotund ( ) Pressure maximum Pressure minimum D 2.0 Volume maximum ( ) Volume minimum Slender (scaled)

V 1.5 STABLE Branch point Cylinder

Volume, 1.0 Region I Region II Region III 0.5 Az

0 0246810 Length, L (scaled)

Figure 7 Stability windows: boundaries of stable regions, in volume-length (VL) space, for a single liquid bridge with pinned CLs for volume (region I) and pressure (region III) disturbances and for a dual-bridge (D) system for a ﬁxed total volume (region II). The instability at boundaries is to rotationally symmetric modes except at rotund and slender limits in which azimuthal (Az) modes are most dangerous. Rendered shapes along boundaries are those before instability. A shape at the Steiner limit, azimuthally unstable, is shown in the upper left. Lengths are scaled by pinning CL radius, and volumes by the cylindrical volume.

positioned above. Activation occurs by an acoustic pulse applied to the lower chamber. In part iii, the up state is overshot, and a liquid bridge forms momentarily until, on rebound (part iv), it breaks, leaving a droplet above. In this way, the bridge grabs the top plate and then releases it. Vogel & Steen (2010) reported an electronically controlled adhesion device based on this grab- and-release action. The device arranges hundreds of droplet switches to act in parallel, toggled by electro-osmotic activation. Bistability is crucial to the design in that no power is needed to maintain either the grabbed or released state, only to toggle between. In another application, in which the pinning of droplets is essential, Lopez´ et al. (2005) tested a variable focal-point lens that dynamically varies the curvatures of the droplets in the switch. Annu. Rev. Fluid Mech. 2015.47:539-568. Downloaded from www.annualreviews.org 2.6. Stability Window: Singlet and Dual Bridges Access provided by Northwestern University - Evanston Campus on 01/06/15. For personal use only. A stability window can be constructed from response diagrams. For the singlet bridge, we start with the TPs in Figure 5. Figure 7 plots the locus of the three TPs, the pressure maximum and minimum and the volume minimum, as L varies. The pressure limits form a closed loop, yielding Axisymmetric mode or shape: mode or the window of stability to pressure disturbances. The window to volume disturbances comprises disturbance shape that the volume limit (by PM theory), supplemented by the branch point curve (large lengths), possesses rotational and the slender and rotund limits for small lengths. The slender and rotund limits are to

symmetry about a azimuthal disturbances (Az), which occur at the Steiner limit and are therefore not picked up generating axis by the PM response diagram, which assumes rotational symmetry. In summary, the full stability window for the pinned singlet bridge, to both pressure and volume disturbances, can be obtained using PM theory to locate axisymmetric instabilities, tracing the TPs in the secondary parameter

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L and combining rotationally symmetric limits with the Steiner limits (Lowry & Steen 1995). The rotationally symmetric limits in Figure 7 were ﬁrst obtained by solving Equation 10 using the conjugate point method (Hormann 1887, Howe 1887) to which the Steiner limits were added (Gillette & Dyson 1971). More recently, Slobozhanin et al. (2012) obtained the bifurcation structures along the edges of the stability window, computing with a weakly nonlinear extension of Equation 10. The stability window for a dual-bridge system can be obtained using PM theory in much the same way. The dual-bridge pressure-volume response is easily constructed graphically

from the singlet response (Figure 5), as V ≡ V 1 + V 2, and equilibrium requires p1 = p2 ≡ p. That is, the system response will have a trunk branch that is identical to the element response but with twice the volume. These states are identical twins. From that trunk, additional branches (non- twins) will bifurcate into the fold direction at each pressure TP. This occurs because, whenever

the response in Figure 5 is triple-valued, there are three choices for V1 and V2 for every p. A conventional bifurcation diagram can be obtained by replotting the response diagram as

V2 − V1 against V, in which bifurcations are seen to be pitchforks. PM theory can be used to ﬁnd the relative stability of each branch. TPs are obtained by introducing an imperfection to break the bifurcations. Applying the PM rules to the imperfect bifurcation yields the set of stability changes. The response can then be deformed back to the perfect state, a deformation that preserves stability changes (Lowry & Steen 1995). The relative stability of all branches is thereby fully resolved. From these, the stability window can then be assembled, as described for the singlet bridge. The resulting dual-bridge limits consist of pressure-minimum, volume-maximum, and rotund curves (Figure 7), extending the single-bridge pressure stability window in the direction of larger volumes. In summary, three stability windows are nested in Figure 7. The pinned singlet bridge subject to volume disturbances, having the largest window, is the most stable; the dual bridge to volume disturbances is the next most stable; and the singlet bridge to pressure disturbances, having the innermost window, is the least stable. Coupling the bridges makes them behave like pressure- disturbed singlet bridges subject to some stabilizing inﬂuence by the system’s volume constraint. This stabilization can be traced back to the birth, on coupling, of the nontwin branch at the maximum pressure TP, most clearly seen in the graphical PM approach.

3. HYDRODYNAMIC STABILITY: A PARALLEL FORMALISM In this section, we establish the parallels between the static and hydrodynamic formulations. For the dynamic formulation, one must solve for the disturbance velocity v and pressure p fields λ Annu. Rev. Fluid Mech. 2015.47:539-568. Downloaded from www.annualreviews.org everywhere in the fluid domain. Normal modes ei t are invoked, which reduces the variables to time independent, and the flow problem (interior domain) is mapped onto the undisturbed

Access provided by Northwestern University - Evanston Campus on 01/06/15. For personal use only. interface using Green’s functions, a boundary-integral approach. Here the frequency λ is scaled by a capillary timescale ρl 3/σ , with ρ the fluid density and l the characteristic length scale. For brevity, we assume that the underlying fluid is inviscid and define the velocity field v =∇φ through the reduced velocity potential φ for irrotational flow. [Inviscid analyses and extensions accounting for viscous effects and vorticity are reported for liquid bridge vibrations (Borkar & Tsamopoulos 1991, Kidambi 2011) and for belted-sphere oscillations (Bostwick & Steen 2013a,b).] One can then write the linearized hydrodynamic field equations as an eigenvalue problem, ∂φ ∂φ 2 2 2 f − − κ + κ = λ φ [∂ D ], (15) ∂n 1 2 ∂n

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α + Δα

αa Λ α 1

αr

u CL Figure 8

Schematic illustration of contact-line speed uCL versus angle α, with advancing αa and receding αr static contact angles (solid lines). Shown is a model of the continuous-speed condition α = g(uCL)(dotted line) with mobility resistance parameter , having limits that recover the free, = 0, and pinned, =∞, contact-line conditions. Figure after Davis (1980).

which is augmented with a CL boundary condition on γ and auxiliary conditions as follows: ∂φ ∂φ ∇2φ = 0[D], ∇φ · n = 0[∂ Ds ], = iλy [∂ D f ], d = 0[∂ D f ]. (16) 1 ∂n ∂n

In other words, the reduced velocity potential must satisfy Laplace’s equation on the ﬂuid domain D, the no-penetration condition on the solid support ∂ Ds , a kinematic condition on free surface ∂ D f that relates the velocity ﬁeld to the interface disturbance there (see Figure 4), and the overall volume constraint. We note the parallel structure of Equations 10 and 15. In the static case (Equation 10), λ<0 corresponds to unstable directions (modes) in the function space, whereas, in the hydrodynamic case (Equation 15), λ2 > 0 corresponds to oscillations. For oscillatory instability, higher modes can be excited and observed. For static stability, special control is needed to observe a higher unstable mode (Thiessen et al. 2002). Typically, only the most unstable mode can be observed. As in the case of static stability, the disturbance behavior at the CL greatly affects dynamic

stability. CL motion is accommodated by a constitutive law, α = g(uCL), that relates the contact Annu. Rev. Fluid Mech. 2015.47:539-568. Downloaded from www.annualreviews.org angle deviation α smoothly to the CL speed uCL as hysteresis is not compatible with the linear

analysis (Figure 8). Linearizing about the static base state uCL = 0 + (∂φ/∂n) gives the following Access provided by Northwestern University - Evanston Campus on 01/06/15. For personal use only. generalization of Equation 14, ∂φ ∂φ ∂φ + χ = iλ [γ ], (17) ∂n ∂n ∂n

with differentiation with respect to the arc-length coordinate, = d/ds. This CL condition was introduced by Davis (1980), although it is sometimes referred to as the Hocking condition, even though Hocking (1987) attributed it to Davis. Here ≡ g(0) is a mobility resistance parameter (see Figure 8). This parameter can be used to smoothly change the boundary conditions between fully mobile (Equation 14), = 0, and pinned (Equation 13), =∞, CLs. The intent here is a phenomenological means to account for CL motion, not a molecular-based model. One

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can think of as a tuning parameter to interpolate between extremes. Reviews of moving CLs include Dussan V. (1979), de Gennes (1985), and Snoeijer & Andreotti (2013).

3.1. Operator Equation and Disturbance Energy To unify the formalism, we recast the eigenvalue problem given by Equation 15 as an operator equation, ∂φ ∂φ λ2 M = K [∂ D f ]. (18) ∂n ∂n Here M is an integral operator representative of the fluid inertia, and K is a differential operator related to the curvature: ∂φ ∂φ ∂φ ∂φ 2 2 M ≡ φ, K ≡− − (κ + κ ) . (19) ∂n ∂n ∂n 1 2 ∂n To proceed with this formulation, one must construct a sufficiently general solution to the boundary value problem, where the volume constraint is incorporated into the Green’s function for the inverse of K, ∂φ ∇2φ = 0[D], = f [∂ D f ]. (20) ∂n k More specifically, given a surface deformation fk, one needs to compute the corresponding velocity potential φk, in accordance with the inertia operator M. For this reason, capillary instability flows are referred to as interfacially driven flows. , ≡ ∂φ/∂ Taking the inner product, ( f g) fgd , of Equation 18 with n yields the disturbance energy equation, ∂φ ∂φ ∂φ ∂φ λ2 M , = K , [∂ D f ]. (21) ∂n ∂n ∂n ∂n Minimizers of the functional L[y] ≡ (K [y], y)/(M [y], y) (22) are necessarily solutions of the governing equations (Equation 15), provided the auxiliary conditions (Equation 16) are satisfied (Myshkis et al. 1987). The no-penetration constraint can be either (a) built into the function space or (b) introduced to the disturbance energy via Lagrange multipliers. The spectrum can be computed via a Galerkin projection of the operator equation (Equation 18) or by the Rayleigh-Ritz method. In the latter, one applies a solution series, = N y j=1 c j y j with yj basis functions, to the functional and minimizes with respect to the coeffi- cients cj to obtain a set of algebraic equations. Bostwick & Steen (2013a) restricted yj, illustrating

Annu. Rev. Fluid Mech. 2015.47:539-568. Downloaded from www.annualreviews.org method (a), whereas Bauer & Chiba (2004) and Vejrazka et al. (2013) used Lagrange multipliers, illustrating method (b). Access provided by Northwestern University - Evanston Campus on 01/06/15. For personal use only.

3.2. Spectral Ordering and Modal Monotonicity: Relative Stability Eigenvalue solutions to self-adjoint operator equations, such as Equation 18, vary with problem parameters. This variation can often be predicted a priori. Here, we paraphrase relevant theorems (Courant & Hilbert 1953) and their implications, restricting the discussion to volume disturbances with pinned ( =∞) or mobile ( = 0) CL conditions, the limiting cases of Equation 17. These make precise the rule of thumb that constraint increases frequency, for example. In view of the parallel formalism, these results hold for both static (Equation 10), with M replaced by the identity operator I in Equation 22, and hydrodynamic (Equation 18) problems.

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2.5 30 m = 6 ab 2.0 6 SSTABLETABLE

20 4 1.5 q λm ,0 PR limit 4 1.0 10 2 Region I 0.5 Region II UNSTABLE 2 0 0 50 70 90 110 130 0 45 90 135 180 α (°) α (°)

Figure 9

Spectral ordering and modal monotonicity. (a) Sessile drop natural frequency λm,0 for the ﬁrst three axisymmetric modes (polar wave numbers m = 2, 4, and 6) against contact angle α for free, F, and pinned, P, disturbances. Circles mark Rayleigh-Lamb frequencies. (b) Static rivulet stability window, axial wave number q against contact angle α, for free (region II), F, and pinned (region I), P, disturbances. For free disturbances, the stability window widens with α (narrows with cosα). The Plateau-Rayleigh (PR) limit ( gray dashed line) is shown for reference. For both the rivulet and drop, pinning stabilizes, consistent with spectral ordering.

3.2.1. Spectral ordering. Let λn and μn be the n-th eigenvalue solutions of Equation 10 or 18 for the pinned, y = 0, and fully mobile, ∂y/∂n + χy = 0, CL conditions, respectively. Then, one

obtains μn ≤ λn. If an equilibrium surface is stable to free disturbances, then it must also be stable to pinned disturbances. Equivalently, if a surface is unstable to pinned disturbances, it must also be unstable to free disturbances.

For the static problem, this makes precise the effect of δ Als = 0 on the minimizers of the functional (Equation 2), discussed in Section 1. Examples include the nested stability windows for the static rivulet (Davis 1980) reproduced in Figure 9b. The stability window for the free CL is nested within that for the pinned CL. For the dynamic problem, this is illustrated in the oscillatory spectrum for the sessile drop in the case of a hemisphere (Lyubimov et al. 2006) and for the more general case of a spherical cap (Bostwick & Steen 2014) (Figure 9a). Relatedly, among admissible disturbances, those that satisfy the natural boundary conditions (Equation 14) are absolute minimizers of Equation 22. That is, the fully mobile disturbance is the most destabilizing disturbance that a capillary surface can be subjected to. Equivalently, it is Annu. Rev. Fluid Mech. 2015.47:539-568. Downloaded from www.annualreviews.org the least constrained (or most unconstrained) disturbance. This is used to identify the appropriate disturbance class (function space) when solving CL conditions such as Equation 17 that are neither Access provided by Northwestern University - Evanston Campus on 01/06/15. For personal use only. pinned nor mobile (discussed below). Free disturbance: disturbance having a free contact line; also 3.2.2. Modal monotonicity. For the natural boundary condition ∂y/∂n + χy = 0, eigenvalue referred to as a natural solution λn of Equation 10 or 18 only change in the same sense as the function χ, where we recall disturbance in the that χ ≡ (k cot α − k¯/ sin α). Eigenvalues vary monotonically with χ, mode-wise (i.e., for fixed n). calculus of variations For fixed interface shape k,α, a concave solid support (k¯ < 0) is relatively stable to a Pinned disturbance: planar support (k¯ = 0), which is relatively stable to a convex support (k¯ > 0). With regards to disturbance having a fixed or immobile Figure 2, this demonstrates that the liquid bridge configuration shown in Figure 2e is rela- contact line tively stable to that in Figure 2h. Or, regarding the far-left column of the figure, free drop configurations become more stable from the bottom to top. Varying k¯ is a homotopy.

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Another implication is that, for an interface in contact with a planar support k¯ = 0, relative stability can be inferred from the sign of χ = k cot α. For the sessile drop, one obtainsχ = cos α, and eigenvalues decrease with increasing α,asinFigure 9a (Bostwick & Steen 2014).

3.3. Dynamic Contact Line and Davis Dissipation The disturbance energy balance for the CL speed condition (Equation 17) follows similarly to Equation 21: ∂φ ∂φ ∂φ 2 ∂φ ∂φ λ2 M , + iλ dγ − K , = 0[∂ D f ]. (23) ∂n ∂n γ ∂n ∂n ∂n On introducing the CL speed condition, Davis (1980) showed that CL motion is purely dissipative for = 0(seeFigure 8), pointing out the formal similarity of Equation 23 to the characteristic equation of a damped harmonic oscillator. To distinguish from viscous dissipation, we refer to this as Davis dissipation. For the free and pinned disturbances, there is no dissipation (Myshkis et al. 1987, Benilov & Billingham 2011). We note that introducing Equation 17 breaks the self-adjoint property of Equation 21, and λ2 will no longer be necessarily real. Lyubimov et al. (2006) showed that the oscillation frequency for the hemispherical drop increases monotonically with from = 0 (free) to =∞(pinned), whereas the decay rate achieves a maximum at a ﬁnite value of the mobility parameter and tends toward zero in the limiting cases = 0, ∞. Bostwick & Steen (2014) considered the more general spherical-cap drop, recovering the results of Lyubimov et al. (2006) regarding dependence.

3.4. Sessile Drops Rayleigh (1879) solved Equations 15 and 16 to show that a free drop (with being a sphere) oscillates with characteristic frequencies λ2 = − + , = , ,..., m m(m 1)(m 2) m 0 1 (24)

and mode shapes Pm given by the Legendre polynomials. We refer to Equation 24 as the Rayleigh spectrum. Lamb (1932) noted that the Rayleigh spectrum is degenerate. For every m > 0, there l λ2 = − + are distinct spherical harmonics mode shapes Y m, with the same frequency m,l m(m 1)(m 2) Rayleigh spectrum: ≤ = l with l m, which we call the Rayleigh-Lamb (RL) spectrum. Azimuthal (l 0) mode shapes Y m the frequencies 0 ≡ are not included in the axisymmetric Rayleigh modes Y m Pm. (Equation 24) at which Sessile drops with α = 90◦ and mobile CLs ( = 0) have RL modes provided that the sum a free capillary drop of m + l is even as these mirror-symmetric disturbances satisfy the no-penetration condition on oscillates, with Annu. Rev. Fluid Mech. 2015.47:539-568. Downloaded from www.annualreviews.org corresponding mode the mid-plane (Equation 16). These modes of course inherit the azimuthal degeneracy of the RL shapes given by the spectrum. Pinning raises the frequency of these sessile drops (see Figure 9a), consistent with Legendre polynomials Access provided by Northwestern University - Evanston Campus on 01/06/15. For personal use only. spectral ordering (see Section 3.2.1). Bostwick & Steen (2014) extended the results of Lyubimov Rayleigh-Lamb (RL) et al. (2006) to both sub- and superhemispherical base states. Basaran & DePaoli (1994) previously spectrum: the calculated the frequency of the lowest-wave-number axisymmetric mode (an extension of the [m, frequencies (Equation l] = [2, 0] RL mode) for pinned drops as it varies with α, using ﬁnite-element simulations. Recent 24) at which a free predictions are in agreement with these prior computations. drop oscillates, with additional azimuthal For a hemispherical drop, Lyubimov et al. (2004) showed that the spectral degeneracy is broken mode shapes given by by varying the mobility parameter = 0. Frequencies split for modes with ﬁxed m, increasing with the spherical the azimuthal wave number l. Asymmetric modes have been observed on gradient surfaces (Daniel harmonics et al. 2004), air cushions (Noblin et al. 2005), and pressure-driven drops (Sharp et al. 2011). Recent observations by Chang et al. (2013) of mechanically oscillated, sessile water droplets catalogued

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PERIODIC TABLE OF MODE SHAPES

The nature of the eigenvalue problem for a sessile drop (Equation 15) invites a comparison to the classic symmetry- breaking properties of the Schrodinger¨ equation. The Schrodinger¨ equation is an eigenvalue problem with spherical symmetry for the hydrogen atom (s orbitals) that gets broken systematically with higher atoms having less symmetry ( p, d,andf orbitals). This symmetry breaking leads to splitting of the eigenvalues (splitting of spectral lines). For a sessile drop, Equation 15 also starts with spherical symmetry (the Rayleigh spectrum) and has subsequent splitting as the support plane is introduced. The drop degeneracy may be broken by (a) the spreading parameter and (b) the base-state volume via the static contact angle α (Bostwick & Steen 2014). For certain values of these symmetry-breaking parameters, two modes may share the same characteristic frequency, or the classical ordering of modes can be altered. To organize and explain the hierarchy of frequencies, one can construct a corresponding periodic table of mode shapes using an Aufbau principle, in which modes are ﬁlled in order of increasing frequency.

the ﬁrst 37 modes arising from the spectral splitting. The [5, 5] mode is shown in Figure 6d.In summary, the RL spectrum, α = 90◦, is found to be largely inadequate as an approximation to the α = 90◦ spectrum for droplets with =∞. In fact, for large deviations from α = 90◦, we showed that spectral reordering may occur (Bostwick & Steen 2014) (see the sidebar Periodic Table of Mode Shapes).

3.4.1. Low-frequency motions and constraint: the Noether mode. The m = 1 mode of the Rayleigh spectrum is a zero-frequency mode related to translational invariance. Translational invariance ensures a first integral of the motion by Noether’s theorem. Accordingly, we refer to m = 1 modes as Noether modes (either [m, l] = [1, 0] or [m, l] = [1, 1], as admissible). Strani & Sabetta (1984) broke the translational invariance by constraining the free drop to contact a spherical-bowl support (Figure 2a). The constraint introduces a new low-frequency oscillatory mode, related to the zero-frequency [1, 0] Noether mode of the Rayleigh spectrum, that correlates with center-of-mass motion. The low-frequency prediction compares well with experiments by Bisch et al. (1982). Similar to the bowl support, a pinned circle of contact also generally introduces a low-frequency mode linked to the Noether mode. Bostwick & Steen (2009) reported that the center-of-mass motion is partitioned among all the eigenmodes, but the low-frequency mode is its principal carrier. If the support circle is placed exactly on a nodal line of a spherical drop, there is of course no influence on the frequency. Otherwise, there is an influence. Ramalingam et al. (2012) and Prosperetti (2012) studied the same problem, both essentially using a Lagrange multiplier Annu. Rev. Fluid Mech. 2015.47:539-568. Downloaded from www.annualreviews.org method to enforce the no-penetration condition at the pinning circle. Ramalingam et al. (2012) explicitly used Lagrange multipliers, whereas Prosperetti (2012) introduced a singular pressure, Access provided by Northwestern University - Evanston Campus on 01/06/15. For personal use only. whose coefficient is determined from the no-penetration condition. Both methods allow for a kink (a discontinuous derivative) in the interface shape at the circle of contact. Bostwick & Steen (2013a) made this disturbance class explicit and showed that a mode with the discontinuous derivative always has a lower frequency than that with a continuous derivative. If the disturbance class (function space) is overly narrow, as when kinks are not allowed at pinning sites, then the frequency will be overpredicted. Increasing the extent of support from a pinning circle to a spherical belt increases the natural frequency, mode-wise, provided the location of the initial pinning circle remains fixed (Bauer & Chiba 2004, Bostwick & Steen 2013a). Yet another drop constraint is related to the Noether mode. One can obtain coupled spherical-cap surfaces by introducing a spherical belt (Figure 2b), centered on the equator with

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the bowl removed. This is related to the capillary switch, discussed above, shown in part i of Figure 6b. Theisen et al. (2007) assumed spherical-cap shapes to model the large-amplitude natural oscillations of the center of mass for the coupled droplet system. Ramalingam & Basaran (2010) employed finite-element simulations to determine the time-dependent response (frequencies, flow fields) of the double-droplet system to (a) pressure forcing and (b) solid substrate oscillations. The low-frequency mode, a relative of the Noether mode, is featured in these studies.

3.4.2. Mobile contact line. Lyubimov et al. (2006) applied the speed condition (Equation 17) to the hemispherical (α = 90◦) base state and reported damped oscillations for 0 <<∞. They showed that the lower-wave-number modes dissipate more energy over an oscillation cycle because they have larger CL displacement, consistent with Equation 23. Fayzrakhmanova & Straube (2009) considered a hemispherical drop subject to forced oscillations. They applied a piecewise speed condition, which admits ﬁnite contact-angle hysteresis, and considered a small- displacement limit to capture stick-slip behavior at intermediate values of . Stick-slip behavior is central to droplet-transport experiments that exhibit ratcheting motion (Daniel et al. 2004; Noblin et al. 2004, 2009). Bostwick & Steen (2014) varied both the contact angle α and the mobility of the CL. Their results compare favorably to experiments (Sharp et al. 2011, Sharp 2012, Chang et al. 2013). They predicted how the mobility will affect the spectrum of the vibrated pinned sessile drops. For piezoelectrically driven drops, Vukasinovic et al. (2007) reported an azimuthal instability, a transition to nonaxisymmetry, related to CL depinning, as suggested in Figure 6c. In addition to affecting the spectrum, CL constraints such as those employed by Mampallil et al. (2011) can enhance mixing in oscillating droplets. Electrowetting techniques have similarly been used to induce CL motion to amplify ﬂow and enhance mixing (Ko et al. 2008, Oh et al. 2008).

3.4.3. Walking drop instability. In view of λ2 > 0 for the Rayleigh spectrum, all sessile-drop motions might be expected to be oscillatory. However, a sessile drop with a fully mobile CL is unstable to the Noether [1, 1] mode for α>90◦,which might have been anticipated by modal monotonicity (Section 3.2.2). We showed that the walking instability occurs by a decrease in both

surface areas, Als and Alg (Bostwick & Steen 2014). Walking correlates with a horizontal center- of-mass motion. That an azimuthal mode can lower the overall surface energy may be surprising. However, in a recent study of the static stability of a free droplet sitting atop the flat end of a supporting rod, Muralidharan et al. (2013) reported an azimuthal shape that yields a lower overall energy A (Equation 2). Annu. Rev. Fluid Mech. 2015.47:539-568. Downloaded from www.annualreviews.org 3.5. Bridge Vibrations Access provided by Northwestern University - Evanston Campus on 01/06/15. For personal use only. Liquid bridge oscillations, natural and forced, parametric and otherwise, have been of interest to the microgravity science community for some time. Interest in float-zone processing of materials in low gravity initially stimulated the activity. The influence of thermocapillarity on bridge stability also received considerable attention. Stability analyses and low-gravity experiments through the 1980s are nicely summarized by Langbein (2002, chapter 12). Since then, there has been considerable progress in solving the stability problem for quiescent base states of cylindrical shape. Much of the focus has been on viscous oscillations, usually with gravity small or neglected. Tsamopoulos et al. (1992) restricted the focus to axisymmetric disturbances and reported resonant frequencies and damping rates for a range of moderate to high Reynolds numbers. Chen & Tsamopoulos (1993) then reported the influence of nonlinearity for forced and free vibrations, and Mollot et al.

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(1993) compared prediction to experiment. Finally, Kidambi (2011) included the azimuthal disturbances to complement Tsamopoulos et al.’s (1992) earlier study. All these modeling studies assume pinned CLs on the end supports. From the viewpoint of this review, the most interesting results emerge from the mobile CL paper that begins this series. Borkar & Tsamopoulos (1991) solved the linear inviscid problem, essentially Equation 15 using the speed condition in Equation 17 controlled by a mobility parameter. They supplemented their inviscid solution with corrections for the viscous boundary layers that occur near the end supports and the weaker ones that occur near the free surface. They observed that (a) as the CL behavior varies between being pinned and free, the damping peaks near the location at which the frequency steps down from its constrained to unconstrained limit; (b) the damping due to Davis dissipation at the CL is more signiﬁcant than that due to viscosity at high-enough Reynolds numbers; and (c) damping is negative for a range of mobilities. Observation (a) is consistent with the mobile drop behavior discussed above. Moreover, observation (b) suggests the importance of Davis dissipation. Finally, observation (c) is suggestive of the instability growth associated with the walking drop instability and consistent with reports by Hocking (1987). Borkar & Tsamopoulos (1991) struggled to interpret the negative damping and suggested that it deserved further study. Pinned CLs were used in all their subsequent studies.

3.6. Rivulets A rivulet is a constrained liquid cylinder in much the same way that a sessile drop is a constrained spherical drop. Starting with a free cylinder, let us imagine just touching a planar support to the cylinder so that a line of contact is established. Davis (1980) showed that this simple constraint

stabilizes the unconstrained cylinder (the PR limit). In terms of an axial wave number, if qc = 1 1/2 corresponds to the PR limit, then qc = (3/4) is the new limit with the line-of-contact support. If the support plane intersects the cylinder at a contact angle α along the two straight parallel CLs, assumed pinned, then the stabilization of the constraint is given by 2 = − π 2/ α 2. qc 1 (2 ) (25) This stability boundary is plotted in Figure 9b. This pinned result was anticipated by Brown & Scriven (1980a), who evaluated Equation 10 directly, and was corroborated by Bostwick & Steen (2010), who recovered the limit from a hydrodynamic analysis for a rivulet constrained by a cylindrical-cup support. Closely related is the meniscus stability on a variety of support geometries for pinned disturbances. There is a large number of studies on cylindrical interfaces. For example, Langbein (1990) studied the interior or exterior wetting of a V-groove, Roy & Schwartz (1999) analyzed a number of cross-sectional containers (e.g., planar, V-groove, circular, and elliptical), Annu. Rev. Fluid Mech. 2015.47:539-568. Downloaded from www.annualreviews.org and Benilov (2009) used lubrication theory to treat a pendant rivulet. In each case, the critical disturbance is the axisymmetric varicose mode. Access provided by Northwestern University - Evanston Campus on 01/06/15. For personal use only. In contrast, mobile CLs are destabilizing to the static rivulet, consistent with the discussion in Section 3.2.2. Figure 9b shows that for α<90◦, shorter rivulets can be destabilized; that is, the stability window shrinks relative to the free cylinder (Davis 1980). We solved the hydrodynamic stability problem for a static rivulet with mobile CLs. Although the varicose mode is the dominant mode, a sinuous (asymmetric) instability also exists, growing at a slower rate. A rivulet with an axial base flow is susceptible to kinematic-wave instabilities characteristic of thin-film flows, especially if the rivulet is relatively flat. Weiland & Davis (1981) and Young & Davis (1987) studied the long-wavelength varicose instabilities of a rivulet with unidirectional gravity-driven flow down a vertical plane and reported capillary instability for narrow rivulets and kinematic-wave instabilities for wide rivulets. Rivulet meandering (sinuous instability) is not as

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well understood as capillary breakup of rivulets and has been the subject of several experimental studies (Schmuki & Laso 1990, Nakagawa & Scott 1992, Nakagawa & Nakagawa 1996). Culkin & Davis (1983) derived a stability index to measure the stabilizing effects of surface tension and the destabilizing effects of inertia under dynamic wetting conditions. Their stability index was only marginally effective at capturing the observed meandering instability, partly because of the inability to model contact-angle hysteresis. Kim et al. (2004) studied a rivulet with a plug-flow base state and used a perturbation analysis to capture the meandering instability. By balancing pressures at the CL, they found a dispersion relation that depends on the base-state geometry, a Weber number, and wetting conditions on the CL. Similarly, Grand-Piteira et al. (2006) derived a meandering-rivulet criterion from a force balance on the CL that incorporates contact-angle hysteresis, capillary effects, and inertia from a gravity-driven base flow. Among other results, they found that the base flow is also hysteretic, and thus the shape of the meandering rivulet varies only with increasing flow rate. In each case, inertia competes with wetting and CL mobility to influence the character of the dominant instability.

3.7. Capillary-Gravity and Faraday Waves Closely related to drops, rivulets and bridges are liquids having a planar free surface, contained laterally. The stability of these capillary surfaces belongs to the study of capillary-gravity (Lamb 1932) and Faraday (1831) waves. For details, we refer the reader to the reviews by Miles & Henderson (1990) and Perlin & Schultz (2000), noting that many of the methods (Benjamin & Scott 1979, Graham-Eagle 1983) and experimental results (Henderson & Miles 1994) parallel those discussed in this review. In some studies of Faraday wave experiments, the threshold acceleration depends strongly on the CL mobility (Nguyem-Thu-Lam & Caps 2011), whereas, in other studies, the presence of a CL has no effect on the modal structure. For example, Edwards & Fauve (1994) demonstrated the existence of pure Faraday waves in irregular-shaped containers. Perlin & Schultz (2000) suggested that improvements to CL models are necessary for additional progress in low-mode Faraday waves, which could explain why damping rates measured experimentally are much larger than the predictions. This comment parallels many of the results mentioned in this review in which CL motion dramatically inﬂuences the instability dynamics.

4. CONCLUDING REMARKS Stability depends on the disturbance class. Going from volume to pressure disturbances destabilizes, whereas going from unpinned to pinned CL stabilizes. Figure 10 summarizes this competition. The pressure disturbance with free CLs is the most dangerous, whereas the Annu. Rev. Fluid Mech. 2015.47:539-568. Downloaded from www.annualreviews.org volume disturbance with pinned CLs is the least dangerous. These results hold for static and dynamic stability and, indeed, for any comparison for which disturbance classes are nested. For Access provided by Northwestern University - Evanston Campus on 01/06/15. For personal use only. any particular capillary surface, the signature of this competition is nested stability windows, as illustrated in Figure 7 for static bridges and Figure 9b for static rivulets. The Steiner symmetrization construction extends the nesting idea. General disturbances are nested, in the sense of an energy ordering, within the rotationally symmetric disturbance class. Another tool, the Poincare´ TP method, makes use of spectral ordering by parametric variation within a disturbance class. Disturbance classes often do not nest, and symmetry breaking often takes the system outside a disturbance class, in which case these tools are not available. In these cases, direct computation may be necessary. The hydrodynamic stability of an inviscid disturbance is probed by normal modes e iλt . With the normal mode ansatz, the linear stability determination reduces to an eigenvalue problem

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vol-free vol-pin

press-free press-pin

Figure 10 Schematic illustration of the relative stability to the pressure disturbances with free contact lines (press-free) and pinned contact lines (press-pin) and volume disturbances with free contact lines (vol-free) and pinned contact lines (vol-pin). Stabilization is in the direction of the arrows.

λ2 M [y] − K [y] = 0 for the stationary shape disturbance y (Equation 18). Here M is a positive- definite operator representing the inertia (mass), and K is an indefinite operator representing the action of capillarity (spring constant). Constraints can be built into the operator or, alternatively, into the function space defining the disturbance class. Hydrodynamic instabilty is related to static instability through a negative eigenvalue of the self-adjoint eigenvalue problem K [y] = λy because a disturbance y yields static instability. A static disturbance y ∗ with negative eigenvalue λ∗ δ2 ≡ = λ of F[y] √ K [y] y corresponds to a growing dynamical disturbance because eigenvalue Annu. Rev. Fluid Mech. 2015.47:539-568. Downloaded from www.annualreviews.org λ∗∗ =−ia −λ∗ with a > 0 by evaluating Equation 22 at y ∗. For rotationally symmetric disturbances, this eigenvalue problem reduces to an ordinary differential equation, in which case Sturm

Access provided by Northwestern University - Evanston Campus on 01/06/15. For personal use only. comparison and separation theorems can be invoked to yield a priori results, as summarized in the spectral ordering and modal monotonicity sections (Sections 3.2.1 and 3.2.2). We hope that the straightforward implications of the theory prove useful to the practitioner. For example, let us suppose the conﬁgurations in Figure 2a,d,g are in equilibrium (liquids and supports may differ). Surface curvatures can be scaled so that k = sin α for all three, which yields

χ = cos α − k¯/ sin α. Then, because k¯a < k¯d (= 0) < k¯g and in view of modal monotonicity, one can conclude that λa <λd <λg for any eigenvalue solutions λ of the second variation (Equation 10). This means that conﬁguration in Figure 2a is more stable than that in Figure 2d,

which is more stable than that in Figure 2g. Moreover, for Figure 2d, because χd = cos α and since λd is monotonic with χ, relative stabilization increases with wettability, a trend that also

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holds for the sessile-drop frequencies. The ability of the CL to anchor, measured by χ,emerges as an important parameter for organizing families of results.

SUMMARY POINTS 1. For both static and dynamic problems and for all families, pinned-CL volume disturbances are the most stabilizing, whereas free-CL pressure disturbances are the least stabilizing. 2. For both the static and dynamic problems, tracking certain families of problems allows relative stability to be predicted a priori. Families following χ and the Noether mode are noteworthy. 3. For both static and dynamic self-adjoint problems with free CLs, eigenvalues increase with χ. 4. For the dynamic problem, the introduction of the lab frame, for example, to make the Rayleigh drop a sessile drop, breaks the translational invariance. This gives rise to walking instability (growth) and other low-frequency center-of-mass motions (oscillation), both inherited from the zero-frequency Noether mode.

FUTURE ISSUES 1. For CLs with mobility, Davis dissipation scales differently from viscous dissipation. Can these forms of dissipation be distinguished in experiment and in simulation? Can the walking instability be observed? 2. Spectral ordering for sessile drops can be broken for a range of wetting and spreading parameters. Can the metaphors of Schrodinger’s¨ equation and the periodic table be exploited for design purposes for drops as well as for liquid bridges and rivulets? 3. The Poincare´ method builds on the isoperimetric problem in which there is a single constraint (volume). Many problems involve multiple constraints, as when a liquid bridge exerts a force at a variable length, along with a pressure at a variable volume. Can the Poincare´ method be generalized to multiple constraints? 4. Steiner symmetrization applies to surfaces with a ﬁxed wetted area (pinned CLs). Can one extend the symmetrization procedure to account for changes in the wetted area (moving Annu. Rev. Fluid Mech. 2015.47:539-568. Downloaded from www.annualreviews.org CLs)? Access provided by Northwestern University - Evanston Campus on 01/06/15. For personal use only.

DISCLOSURE STATEMENT The authors are not aware of any biases that might be perceived as affecting the objectivity of this review.

ACKNOWLEDGMENTS We thank A. Hirsa, M. Smith, D. Thiessen, H. van Lengerich, and C.T. Chang for supplying some of the experimental images and movies used in this review. P.H.S. thanks the NSF for support by grant CBET1236582.

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Annu. Rev. Fluid Mech. 2015.47:539-568. Downloaded from www.annualreviews.org RELATED RESOURCES Bernate JA, Thiessen DB. 2012. Stability of capillary surfaces supported by a helical wire. APS Access provided by Northwestern University - Evanston Campus on 01/06/15. For personal use only. DFD Gallery Fluid Motion video. http://arxiv.org/abs/1210.4100. Illustrates stabilizing liquid columns by helical supports. Chang C-T, Daniel S, Steen PH. 2013. Excited sessile drops dance harmonically. YouTube video. http://www.youtube.com/watch?v=Bk7xh4DqkiY. Shows a classiﬁcation scheme for the identiﬁcation of mechanically oscillated sessile drops.

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Annual Review of Fluid Mechanics Contents Volume 47, 2015

Fluid Mechanics in Sommerfeld’s School Michael Eckert pppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppp1 Discrete Element Method Simulations for Complex Granular Flows Yu Guo and Jennifer Sinclair Curtis ppppppppppppppppppppppppppppppppppppppppppppppppppppppppp21 Modeling the Rheology of Polymer Melts and Solutions R.G. Larson and Priyanka S. Desai pppppppppppppppppppppppppppppppppppppppppppppppppppppppppp47 Liquid Transfer in Printing Processes: Liquid Bridges with Moving Contact Lines Satish Kumar pppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppp67 Dissipation in Turbulent Flows J. Christos Vassilicos ppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppp95 Floating Versus Sinking Dominic Vella pppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppp115 Langrangian Coherent Structures George Haller pppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppp137 Flows Driven by Libration, Precession, and Tides Michael Le Bars, David C´ebron, and Patrice Le Gal ppppppppppppppppppppppppppppppppppppp163 Fountains in Industry and Nature G.R. Hunt and H.C. Burridge ppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppp195 Acoustic Remote Sensing Annu. Rev. Fluid Mech. 2015.47:539-568. Downloaded from www.annualreviews.org David R. Dowling and Karim G. Sabra ppppppppppppppppppppppppppppppppppppppppppppppppppp221

Access provided by Northwestern University - Evanston Campus on 01/06/15. For personal use only. Coalescence of Drops H. Pirouz Kavehpour ppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppp245 Pilot-Wave Hydrodynamics John W.M. Bush pppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppp269 Ignition, Liftoff, and Extinction of Gaseous Diffusion Flames Amable Liñán, Marcos Vera, and Antonio L. Sánchez pppppppppppppppppppppppppppppppppppp293 The Clinical Assessment of Intraventricular Flows Javier Bermejo, Pablo Mart´ınez-Legazpi, and Juan C. del Alamo´ pppppppppppppppppppppp315

v FL47-FrontMatter ARI 22 November 2014 11:57

Green Algae as Model Organisms for Biological Fluid Dynamics Raymond E. Goldstein ppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppp343 Fluid Mechanics of Blood Clot Formation Aaron L. Fogelson and Keith B. Neeves pppppppppppppppppppppppppppppppppppppppppppppppppppp377 Generation of Microbubbles with Applications to Industry and Medicine Javier Rodr´ıguez-Rodr´ıguez, Alejandro Sevilla, Carlos Mart´ınez-Baz´an, and Jos´e Manuel Gordillo pppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppp405 Beneath Our Feet: Strategies for Locomotion in Granular Media A.E. Hosoi and Daniel I. Goldman ppppppppppppppppppppppppppppppppppppppppppppppppppppppppp431 Sports Ballistics Christophe Clanet pppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppp455 Dynamic Stall in Pitching Airfoils: Aerodynamic Damping and Compressibility Effects Thomas C. Corke and Flint O. Thomas pppppppppppppppppppppppppppppppppppppppppppppppppppp479 Ocean Spray Fabrice Veron pppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppp507 Stability of Constrained Capillary Surfaces J.B. Bostwick and P.H. Steen ppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppp539 Mixing and Transport in Coastal River Plumes Alexander R. Horner-Devine, Robert D. Hetland, and Daniel G. MacDonald ppppppppp569

Indexes

Cumulative Index of Contributing Authors, Volumes 1–47 pppppppppppppppppppppppppppp595 Cumulative Index of Article Titles, Volumes 1–47 ppppppppppppppppppppppppppppppppppppppp605

Errata

Annu. Rev. Fluid Mech. 2015.47:539-568. Downloaded from www.annualreviews.org An online log of corrections to Annual Review of Fluid Mechanics articles may be found at http://www.annualreviews.org/errata/ﬂuid Access provided by Northwestern University - Evanston Campus on 01/06/15. For personal use only.

vi Contents